limit as y approaches infinity of (3+2/(y^3)-3y^4)/(9y^4-5/(y^2)-3)

解答

y \to \infty lim (\frac{3 + \frac{2}{y ^{3}} - 3 y ^{4}}{9 y ^{4} - \frac{5}{y ^{2}} - 3})

- \frac{1}{3}

十进制

- 0.33333 \dots

求解步骤

y \to \infty lim (\frac{3 + \frac{2}{y ^{3}} - 3 y ^{4}}{9 y ^{4} - \frac{5}{y ^{2}} - 3})

乘开

\frac{3 + \frac{2}{y ^{3}} - 3 y ^{4}}{9 y ^{4} - \frac{5}{y ^{2}} - 3} : \frac{3 y ^{3} + 2 - 3 y ^{7}}{9 y ^{7} - 3 y ^{3} - 5 y}

= y \to \infty lim (\frac{3 y ^{3} + 2 - 3 y ^{7}}{9 y ^{7} - 3 y ^{3} - 5 y})

除以分母的最高次幂:

\frac{\frac{3}{y ^{4}} + \frac{2}{y ^{7}} - 3}{9 - \frac{3}{y ^{4}} - \frac{5}{y ^{6}}}

= y \to \infty lim (\frac{\frac{3}{y ^{4}} + \frac{2}{y ^{7}} - 3}{9 - \frac{3}{y ^{4}} - \frac{5}{y ^{6}}})

x \to a lim [\frac{f ( x )}{g ( x )}] = \frac{lim _{x \to a} f ( x )}{lim _{x \to a} g ( x )}, x \to a lim g (x) \neq = 0

不定型除外

= \frac{lim _{y \to \infty} ( \frac{3}{y ^{4}} + \frac{2}{y ^{7}} - 3 )}{lim _{y \to \infty} ( 9 - \frac{3}{y ^{4}} - \frac{5}{y ^{6}} )}

y \to \infty lim (\frac{3}{y ^{4}} + \frac{2}{y ^{7}} - 3) = - 3

y \to \infty lim (9 - \frac{3}{y ^{4}} - \frac{5}{y ^{6}}) = 9

= \frac{- 3}{9}

化简

\frac{- 3}{9} : - \frac{1}{3}

= - \frac{1}{3}

x \to \infty lim (\frac{3 x + 1}{4 x ^{2} - 3 x - 7})

x \to 4 lim (x^{2} + 3 x - 4)

x \to 0 lim (4 x^{7 x})

x \to 0 lim (ln (x^{x}))

x \to - \infty lim (x, 4)